Large-eddy simulation of multiphase flows incomplex combustors

S. V. Apte1, K. Mahesh2, F. Ham1, G. Iaccarino1, G.Constantinescu1, & P. Moin1

1 Center for Integrated Turbulence Simulations, Stanford University2 University of Minnesota

Abstract

Large-eddy simulation (LES) is a promising technique to accurately predictreacting multi-phase flows in practical combustors involving complex physicalphenomena of turbulent mixing and combustion dynamics. Our goal in the presentwork is to develop a computational tool based on particle-tracking schemes capa-ble of performing hi-fidelity multiphase flow simulations with models to captureliquid-sheet breakup, droplet evaporation, droplet deformation and drag. An Eule-rian low-Mach number formulation on arbitrary shaped unstructured grids is usedto compute the gaseous phase. The dispersed phase is solved in a Lagrangianframework by tracking a large number of particles on the unstructured grid. Theinterphase mass, momentum, and energy transport are modeled using two-waycoupling of point-particles. A series of validation simulations are performed incoaxial and realistic gas-turbine combustor geometries to evaluate the predictionsmade for multiphase, turbulent flow.

1 Introduction

Turbulent multi-phase flows are encountered in a variety of engineering applica-tions, viz., internal combustion engines, liquid and solid propellant rocket motors,gas-turbine aircraft engines, cyclone combustors and biomass gasifiers involv-ing swirling motions. The combustion chambers of propulsion systems involveintriguing processes such as break-up and atomization of liquid fuel jets, i evapo-ration/condensation and collision/coalescence of droplets, turbulent mixing of fueland oxidizer giving rise to spray-flames. Owing to the complexities of the under-lying processes, accurate quantitative observations of the flowfield are formidable

and often subject to large measurement errors. Better understanding of such flowsfor design modifications, improvements, and exploring fundamental physical phe-nomena demands high-fidelity numerical studies in realistic configurations.

To date the engineering prediction of such flows in realistic configurations reliespredominantly on the Reynolds-averaged Navier-Stokes equations (RANS). Thelarge-eddy simulation (LES) technique has been convincingly shown to be superiorto RANS in accurately predicting turbulent mixing and combustion dynamics [1]in simpler combustor geometries. Recently, Mahesh et al. [2] have developed anovel numerical algorithm for high-fidelity turbulence simulations on unstructuredgrids and complex geometries. This solver is extended to perform multiphase flowsimulations with Lagrangian particle tracking on unstructured grids [3, 4]. Theparticles are treated as point-sources with two-way coupling of mass, momentum,and energy between the gas and dispersed phases.

The paper is organized as follows. Section 2 provides a brief overview of the gasand particle phase equations. The droplet evaporation and breakup models are thendescribed. Section 3 presents results from the LES of various test cases performedin simple coaxial as well as complex realistic gas-turbine combustor geometries.

2 Theoretical Formulation

The governing equations used for the gaseous and dispersed phases are brieflydescribed.

2.1 Gas-Phase

The three-dimensional, low-Mach number, filtered Navier-Stokes equations aresolved on unstructured grids with arbitrary elements. These equations are writtenas

∂ρguj∂xj

= −∂ρg∂t

+ Sm (1)

∂ρgZ

∂t+∂ρgZuj∂xj

=∂

∂xj(ρgαZ

∂Z

∂xj)− ∂qZj

∂xj+ SZ (2)

∂ρgC

∂t+∂ρgCuj∂xj

=∂

∂xj(ρgαC

∂C

∂xj)− ∂qCj

∂xj+ ωC (3)

∂ρgui∂t

+∂ρguiuj∂xj

= − ∂p

∂xi+∂(2µSij)

∂xj− ∂qij∂xj

+ Si (4)

ρg = ρg(Z, C, Z”) (5)

Sij =1

2(∂ui∂uj

+∂uj∂ui

)− 1

3δij∂uk∂xk

(6)

where the unclosed transport terms in the momentum and scalar equations aregrouped into the residual stress qij , and residual scalar flux qZj , qCj . The dynamicSmagorinsky model by Moin et al. [5] is used to close these subgrid terms asdemonstrated by Pierce & Moin [1]. For a two-fluid (air + fuel) mixture, one con-served scalar (the mixture fraction, Z) and a non-conserved scalar (the progressvariable, C) are solved. By using the standard low-Mach number assumption andassuming unity Lewis number, the energy equation can be written as a conservedscalar transport. In addition, assuming adiabatic walls the energy equation has thesame boundary condition as the mixture fraction, and the two are linearly depen-dent. The density is obtained from the lookup tables generated using the flamelettheory for non-premixed combustion and is dependent on the local values of thetwo scalars and the subgrid mixture fraction fluctuations, Z”. The source term,ωC , represents the reaction source term obtained from flamelet theory (Pierce &Moin [1]). The cooling effect due to evaporation of droplets (heat of vaporization)can be accounted for in the equation of state by appropriately reducing the density.The density field is then obtained as a function of the mixture fraction, fluctua-tions in mixture fraction, and the progress variable from lookup tables generatedusing the flamelet theory for non-premixed combustion [1]. The source terms inthe continuity, momentum, and scalar transport equations are due to the interphaseinteractions.

2.2 Particle-Phase

The particle motion is simulated using the Basset-Boussinesq-Oseen (BBO) equa-tions [6]. It is assumed that the density of the particle is much larger than that ofthe fluid (ρp/ρf ∼ 103), particle-size is small compared to the turbulence integrallength scale, and that the effect of shear on particle motion is negligible. The highvalue of ρp/ρf implies that the Basset force and the added mass term are smalland are therefore neglected. Under these assumptions, the Lagrangian equationsgoverning the particle motions in non-dimensional form are

dxpdt

= up, (7)

dupdt

=

(1 + aRebp

)118ρpd

2pReref

(u− up) +LrefU2ref

g (8)

where Rep = dpReref |u − up| is the particle Reynolds number and Reref isthe reference Reynolds number based on a reference length, velocity and densityLref , Uref , ρref , respectively. For particle Reynolds number up to 800, the con-

stants used in the drag law are a = 0.15, b = 0.687 [6]. The source-term Si inthe momentum-equations represent the ‘two-way’ coupling between the gas andparticle-phases and is given by

Si = − 1

Vcv

∑

k

ρkpρref

V kpdukpidt

(9)

where the subscript p stands for the particle phase. Here, Vcv and V kp = π6 d

kp

3are

the volumes of the computational cell and particle k, respectively. The∑k is over

all particles in a computational control volume.The gas-phase velocity, u, in equations (7) and (8) is computed at individual

particle locations within a control volume using a generalized, tri-linear interpo-lation scheme for arbitrary shaped elements. Introducing higher order accurateinterpolation is straight forward; however, it was found that tri-linear interpolationis sufficient to represent the gas-phase velocity field at particle locations.

2.2.1 Atomization and BreakupLiquid spray atomization plays a crucial role in analyzing the combustion dynam-ics in gas-turbine combustors. In order to predict the essential global features ofatomization, a stochastic model for secondary breakup that accounts for a rangeof product-droplet sizes is used. Specifically, for a given control volume, the char-acteristic radius of droplets is assumed to be a time-dependent stochastic variablewith a given initial distribution function. The breakup of parent blobs into sec-ondary droplets is viewed as the temporal and spatial evolution of this distributionfunction around the parent-droplet size. This distribution function follows a cer-tain long-time behavior, which is characterized by the dominant mechanism ofbreakup. The size of new droplets is then sampled from the distribution functionevaluated at a typical breakup time scale of the parent drop as described by Apteet al. [7]. The discrete model by Kolmogorov is reformulated in terms of a Fokker-Planck (FP) differential equation for the evolution of the size-distribution functionfrom a parent-blob towards the log-normal law:

∂T (x, t)

∂t+ ν(ξ)

∂T (x, t)

∂x=

1

2ν(ξ2)

∂2T (x, t)

∂x2. (10)

where the breakup frequency (ν) and time (t) are introduced. Here, T (x, t) is thedistribution function for x = log(rj), and rj is the droplet radius. Breakup occurswhen t > tbreakup = 1/ν. The value of the breakup frequency and the criti-cal radius of breakup are obtained by the balance between the aerodynamic andsurface tension forces. The secondary droplets are sampled from the analyticalsolution of equation (10) corresponding to the breakup time-scale. The parametersencountered in the FP equation (〈ξ〉 and

⟨ξ2⟩) are computed by relating them to

the local Weber number for the parent blob, thereby accounting for the capillaryforces and turbulent properties. As new droplets are formed, parent droplets aredestroyed and Lagrangian tracking in the physical space is continued till furtherbreakup events.

2.2.2 Hybrid Approach for Particle TrackingPerforming spray breakup computations using Lagrangian tracking of each indi-vidual droplet gives rise to a large number of droplets (∼ 50 million) very closeto the injector. A hybrid scheme involving the computation of both individualdroplets and parcels is used [7]. The difference between droplets and parcels issimply the number of particles associated with them, Npar, which is unity for

x/Ry/

R

0 5 10 15-3

-2

-1

0

1

2

3

1.5

1.2

0.8

0.5

0.2

-0.2

-0.5

z/R-2 0 2

-3

-2

-1

0

1

2

3 x/R=2.66

z/R-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

z/R

y/R

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3 x/R=1.63x/R=0.786

z/R=0

Figure 1: Snapshots of particles superposed on contours of instantaneous axialvelocity at different cross-sections. Conditions correspond to the experi-ment by Sommerfeld & Qiu [10].

droplets. During injection, new particles added to the computational domain arepure drops (Npar = 1). These drops move downstream and undergo breakupaccording to the above breakup model and produce new droplets. The basic ideabehind the hybrid-approach, is to collect all droplets in a particular control vol-ume and group them into bins corresponding to their size and other propertiessuch as velocity, temperature etc. The droplets in bins are then used to form a par-cel by conserving mass, momentum, and energy. The properties of the parcel areobtained by mass-weighted averaging from individual droplets in the bin. For thisprocedure, only those control volumes for which the number of droplets increasesabove a certain threshold value are considered. The parcel thus created then under-goes breakup according to the above stochastic model, however, does not createnew parcels. On the other hand, Npar is increased and the diameter is decreasedby mass-conservation.

2.2.3 Droplet EvaporationThe evaporation model is based on the single (isolated) droplet evaporation process(see e.g. Faeth [8]). The droplets are assumed to have homogeneous temperatureand spherical shape. The heat and mass balance equations at the droplet surfaceare used to evaluate droplet temperature (Tp), evaporation rate dmp/dt, dropletmass (mp), and diameter (dp). Multiplicative correction factors account for theconvective effects [8]. Effects of droplet deformation, internal circulation on dragare modeled using the correlations provided by Helenbrook & Edwards [9]. Equa-tions (7) and (8) are integrated using a fourth-order Runge-Kutta time-steppingalgorithm. After obtaining the new particle positions,the particles are relocated,particles that cross interprocessor boundaries are duly transferred, boundary con-

z/R-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

x/R

r/R

0 5 100

1

2

3

dp =100 µm

r/R

-1 0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

dp = 80 µm

r/R

0 5 100

1

2

3

dp = 60 µm

r/R

-1 0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

dp = 30 µm

Figure 2: Particle trajectories for different size classes.

ditions on particles crossing boundaries are applied, source terms in the gas-phaseequation are computed, and the computation is further advanced.

3 Results

A systematic evaluation of the present multiphase solver is performed by perform-ing validation simulations in coaxial and realistic combustor geometries: 1) non-reacting, particle-laden, swirling flow in a coaxial combustor [10], 2) evaporat-ing droplets in a coaxial combustor [11], 3) spray breakup in a cylindrical Dieselengine combustion chamber, 4) atomization and breakup in a realistic gas-turbinecombustor. Some salient features of these simulations are presented next. Detailsof some of these computations can also be found in [3, 4, 7].

3.1 Swirling Particle-Laden Flow

Figure (1) shows the instantaneous contours of axial velocity in four differentcross-sections superimposed by the scatter plot of particle positions correspondingto the experiments of Sommerfeld & Qiu [10]. A mixture of air and lightly loaded,spherical, glass-particles with a prescribed size-distribution enters the primary jet,while a swirling stream of air flows through the annulus. Good agreement with theexperiment is obtained in predicting the mean and rms components of velocity for

X

Y

0 5 10

-2

-1

0

1

2

1.0E-03 1.6E-03 2.6E-03 4.1E-03 6.6E-03 1.0E-02 1.7E-02 2.7E-02 4.3E-02 6.9E-02YF

Figure 3: Snapshots of droplets superposed on contours of instantaneous fuel massfraction. Conditions correspond to the experiment by Sommerfeld &Qiu [11].

both phases as well as the mean and rms diameter of the particles at several axialstations [4]. Each particle injected into the combustion chamber is tracked and astationary number of approximately 1.1 million particles is obtained within thecomputational domain. Statistics of both phases were then collected over a coupleof flow through times to obtain good results.

An in depth analysis of the particle dispersion characteristics is also performedby tracking 20,000 tagged particles of different size class. The trajectories of thedifferent size particles are shown in Fig. (2). It is clearly seen that particles withdiameters less than 80 µm decelerate to zero velocity within the central recircula-tion bubble. Their larger inertia causes large diameter particles to penetrate moreinto the recirculation zone before coming to a halt. These particles then acceleratetowards the inlet and are transported radially outwards by the centrifugal forces.The particles then exit the region of reverse flow and are entrained by the annularjet which transports them downstream with swirling motion and multiple reflec-tions off the wall. The particle residence times, size-velocity correlations obtainedalso show good agreement with the experimental data.

3.2 Droplet Evaporation in a Coaxial Combustor

After obtaining good predictions of the particle-laden, swirling flow a simula-tion of evaporating spray in a coaxial geometry is performed corresponding to theexperiments of Sommerfeld & Qiu [11]. The contours of instantaneous fuel massfraction with scatter plot of droplets superimposed is shown in Fig. (3). Hot air (T= 373 K) is injected through the annulus. Liquid isopropyl alcohol is injected by anozzle mounted at the center of the coaxial combustor. It forms a conical spray andthe size distributions and the droplet size-velocity correlations measured at the inletsection are used as the inlet conditions. The initial temperature of the liquid is 313K and is below its boiling point (355 K). Since, the temperature of the surroundinghot air is low, there are no reactions. Figure (4) shows the comparison of the liquidphase statistics with the experimental data. The droplet size distributions shows atypical conical spray near the injector with large size droplets accumulating on the

Axial Velocity0 10 20

-80

-60

-40

-20

0

20

40

60

80

X/R = 3.125

Axial Velocity0 5 10 15

-80

-60

-40

-20

0

20

40

60

80

X/R = 6.25

Axial Velocity0 10 20

-80

-60

-40

-20

0

20

40

60

80

X/R = 9.375

Axial Velocity0 10 20

-75

-50

-25

0

25

50

75

X/R = 12.5

Mass Flux, kg/m2-s0 200

-80

-60

-40

-20

0

20

40

60

80

X/R = 3.125

Mass Flux, kg/m2-s0 100

-80

-60

-40

-20

0

20

40

60

80

X/R = 3.125

Mass Flux, kg/m2-s0 50

-80

-60

-40

-20

0

20

40

60

80

X/R = 9.375

Mass Flux, kg/m2-s0 20 40 60

-75

-50

-25

0

25

50

75

X/R = 12.5

Axial Velocity0 10 20

-80

-60

-40

-20

0

20

40

60

80

X/R = 1.56

Mass Flux, kg/m2-s0 200 400

-80

-60

-40

-20

0

20

40

60

80

X/R = 1.56

Diameter. µm0 50

-60

-40

-20

0

20

40

60

X/R = 1.56

Diameter, µm0 25 50 75

-60

-40

-20

0

20

40

60

X/R = 3.125

Diameter, µm0 50

-60

-40

-20

0

20

40

60

X/R = 6.25

Diameter, µm0 20 40

-60

-40

-20

0

20

40

60

X/R = 9.375

Diameter, µm0 20 40

-50

-25

0

25

50

X/R = 12.5

Axial Velocity

R,m

m

0 10 20

-75

-50

-25

0

25

50

75

X/R = 0.786

Mass Flux, kg/m2-s

R,m

m

0 500 1000

-75

-50

-25

0

25

50

75

X/R = 0.786

Diameter. µm

R,m

m

0 50 100

-50

-25

0

25

50

X/R = 0.786

Figure 4: Comparison of simulation results with experimental data: mean and rmsof droplet axial velocity, axial mass flux of the liquid fuel, mean and rmsof droplet diameter

outer edge of the spray and small droplets in the core. These droplets evaporateand their mean diameter decreases further downstream. Predictions of the meanand rms of the axial velocity, the mean axial mass flux, and the mean and rms ofdroplet diameter at different axial locations are good compared to the experimentsand show significant improvements over the corresponding RANS predictions.

3.3 Atomization and Spray Evolution in a Realistic Geometry

The stochastic model along with the hybrid particle-parcel approach are used tosimulate spray evolution from a Pratt and Whitney injector. The experimental dataset was obtained by mounting the injector in a cylindrical plenum through whichair with prescribed mass-flow rate was injected. The air goes through the main andguide swirler to create a swirling jet into the atmosphere. Liquid film is injectedthrough the filmer surface which forms an annular ring. The liquid mass-flow ratecorresponds to certain operating conditions of the gas-turbine engine. Experimen-tal data in terms of the droplet mean diameter, size distribution, and liquid massflux in the radial direction at two different axial locations away from the injectorare available. Gas-phase statistics for mean and rms velocities is also available atthese locations. A snapshot of the spray evolution in the z = 0 plane along withthe gas-phase axial velocity contours is shown in Fig.5. The hybrid-approach usedherein gives a dynamical picture with correct spray angle. Results show that theliquid mass fluxes at two downstream locations are in good agreement with the

Figure 5: Instantaneous snapshot of spray evolution from a PW Injector.

experimental data. The droplet size-distribution is also in reasonable agreementconsidering that coalescence effects are neglected. This shows that the presentsolver can handle extremely complex and realistic combustor geometries. A com-plete computation of spray breakup, droplet evaporation, and non-premixed com-bustion in a real PW combustion chamber is being performed to understand thespray flame dynamics.

4 Summary

We have developed a Lagrangian Particle Tracking (LPT) framework for an unstruc-tured grid, arbitrary shape LES solver developed by Mahesh et al. [2, 3]. Thepresent solver can be directly used to accurately compute multi-physics, mul-tiphase flows in realistic gas-turbine combustor geometries. The accuracy androbustness of the solver as well as its predictive capability of complex flows areverified by performing several validation studies of particle-laden, swirling flowsin a coaxial combustor geometry for evaporating liquid and non-evaporating solidparticles. The interphase mass, momentum, and energy exchange is modeled bytwo-way coupling between the phases. A stochastic model developed by Apte etal. [7] is also used to perform simulation of spray atomization in a realistic gas-turbine injector. The results obtained from these simulations are in good agree-ment with the available experimental data. The present solver is being extendedto perform turbulent, reacting, multiphase flow simulations in complex combustorgeometries.

Acknowledgments

Support for this work was provided by the United States Department of Energyunder the Accelerated Strategic Computing Initiative (ASCI), program. The com-puter resources provided on Blue Horizon at San Diego Supercomputing Cen-ter and ASCI Frost at Lawrence Livermore National Laboratory, CA are greatlyappreciated. to Dr. J. C. Oefelein of Sandia National Labs, Dr. G. Iaccarino of theCenter for Turbulence Research for their help at various stages of this study.

References

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[2] Mahesh, K., Constantinescu, G., & Moin, P., A new time-accurate finite-volume fractional-step algorithm for prediction of turbulent flows on unstruc-tured hybrid meshes. in preparation for Journal of Computational Physics.,2003.

[3] Mahesh, K., Constantinescu, G., Apte, S., Iaccarino, G., Ham, F., & Moin,P., Large-eddy simulation of gas turbines combustors. Annual Research Briefs ,Center for Turbulence Research, Stanford University, pp. 3-17, 2002.

[4] Apte, S.V., Mahesh, K., Moin, P., & Oefelein, J.C., Large-eddy simulation ofswirling particle-laden flow in a coaxial-jet combustor, to appear in Interna-tional Journal of Multiphase Flow, 2003.

[5] Moin, P., Squires, K., Cabot, W., & Lee, S., A dynamic subgrid model forcompressible turbulence and scalar transport. Physics of Fluids. A., 3, pp. 2746-2757, 1991.

[6] Crowe, C., Sommerfeld, M., & Tsuji, Y., Multiphase flows with droplets andparticles CRC Press, Boca Raton, FL., 1998.

[7] Apte, S.V., Gorokhovski, M., & Moin, P., LES of atomizing spray withstochastic modeling of secondary breakup. submitted for publication in Inter-national Journal of Multiphase Flow, 2003.

[8] Faeth, G. M., Mixing, transport, and combustion in sprays. Progress in Energyand Combustion Science, 13, pp. 293-345, 1987.

[9] Helenbrook, B. T., & Edwards, C. F., Quasi-steady Deformation and Dragof Uncontaminated Liquid Drops. International Journal of Multiphase Flows,2002.

[10] Sommerfeld, M. & Qiu, H. H., Detailed measurements in a swirling particu-late two -phase flow by a phase - doppler anemometer. International Journal ofHeat and Fluid Flow 12(1), pp. 20 - 28, 1991.

[11] Sommerfeld, M. & Qiu, H. H., Experimental study of spray evaporation inturbulent flow. International Journal of Heat and Fluid Flow 19, pp. 10 - 22,1998.